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4.8.12 Ring Mappings: the Image Function

The function Image implements a ring homomorphism. Suppose S is the current ring and R is another ring. If X is an object in R, the function Image may be used to substitute polynomials in S for the indeterminates in X. An example is given below and complete details are given in the online help entry for Image.

To make substitutions within a single ring, one would usually use Eval or Subst rather than Image. To map a polynomial or ideal from an outside ring into the current ring, the functions QZP and ZPQ are sometimes useful. To map a polynomial or rational function (or a list, matrix, or vector of these) from R to S without changing indeterminates, use the function BringIn. ("BringIn" is only applicable if the indeterminates of the object to be mapped are a subset of those in S.)

example

    
Use R ::= Q[a,b,c];
X := a+b-3c;
Use S ::= Q[x,y]; 
F := RMap(x^2,2,y^2);  -- syntax for defining a map: the n-th
             -- indeterminate in the domain will be mapped to
             -- the n-th element listed in RMap.
X; -- X lives in the ring R
R :: a + b - 3c
-------------------------------
Image(X,F); -- the image of E under the map F
x^2 - 3y^2 + 2
-------------------------------
Image(R:: (a+b)^2,F);
x^4 + 4x^2 + 4
-------------------------------